Universal instantiation

In predicate logic, universal instantiation^[1]^[2]^[3] (UI; also called universal specification or universal elimination, and sometimes confused with dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom schema. It is one of the basic principles used in quantification theory.

Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal."

In symbols, the rule as an axiom schema is

\forall x\,A(x)\Rightarrow A(a/x),

for every formula A and every term a, where $A(a/x)$ is the result of substituting a for each free occurrence of x in A. $\,A(a/x)$ is an instance of $\forall x\,A(x).$

And as a rule of inference it is

from ⊢ ∀x A infer ⊢ A(a/x),

with A(a/x) the same as above.

Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934." ^[4]

Quine

According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that "∀x x = x" implies "Socrates = Socrates", we could as well say that the denial "Socrates ≠ Socrates" implies "∃x x ≠ x". The principle embodied in these two operations is the link between quantifications and the singular statements that are related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term names and, furthermore, occurs referentially.^[5]

References

^ Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010). Introduction to Logic. Pearson Education. ISBN 978-0205820375.^{[page needed]}
^ Hurley^{[full citation needed]}
^ Moore and Parker^{[full citation needed]}
^ Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ
^ Willard Van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality". Quintessence. Cambridge, Mass: Belknap Press of Harvard University Press. OCLC 728954096. Here: p. 366.

[1] Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010). Introduction to Logic. Pearson Education. ISBN 978-0205820375.^{[page needed]}

[2] Hurley^{[full citation needed]}

[3] Moore and Parker^{[full citation needed]}

[4] Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ

[5] Willard Van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality". Quintessence. Cambridge, Mass: Belknap Press of Harvard University Press. OCLC 728954096. Here: p. 366.

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Quine

See also

References